Many humanoid and multi-legged robots are controlled in positions rather than in torques, which prevents direct control of contact forces, and hampers their ability to create multiple contacts to enhance their balance, such as placing a hand on a wall or a handrail. This paper introduces the SEIKO (Sequential Equilibrium Inverse Kinematic Optimization) pipeline, and proposes a unified formulation that exploits an explicit model of flexibility to indirectly control contact forces on traditional position-controlled robots. SEIKO formulates whole-body retargeting from Cartesian commands and admittance control using two quadratic programs solved in real time. Our pipeline is validated with experiments on the real, full-scale humanoid robot Talos in various multi-contact scenarios, including pushing tasks, far-reaching tasks, stair climbing, and stepping on sloped surfaces. Code and videos are available at: https://hucebot.github.io/seiko_controller_website/